LMA - Laboratoire de Mécanique et d’Acoustique

[Confinement / visioconf.] Caroline Bauzet - Existence and uniqueness result for an hyperbolic scalar conservation law with a stochastic force using a finite volume approximation


Le 23 juin 2020 de 11h00 à 12h00

Caroline BAUZET : MC (équipe M&S)
Travail commun avec V. CASTEL, et J. CHARRIER

Keywords : Stochastic PDE, multiplicative noise, finite volume method, monotone scheme, entropy solution, doubling variable method, Young measures, measure-valued entropy solution.


In this talk, I will present a joint work ([BCC20]) on the Cauchy problem for a nonlinear hyperbolic scalar conservation law in d space dimensions forced by a multiplicative stochastic noise (in the sense of Itô) and with a general time and space dependent flux-function of type :

PNG - 269.8 ko

The well-posedness theory for (1) is a known result since the work of [DeV10] by the way of a kinetic approach and the numerical approximation of its solution has been the subject of few work recently ([BCG16, M18]...). Our aim here is to address simultaneously theoretical and numerical issues. More precisely I will present an existence and uniqueness result of the stochastic entropy solution of (1) together with its approximation by a finite volume scheme. Comparing with the existing results on the subject, the true novelty of the present study is the use of the numerical approximation to get both existence and uniqueness of the stochastic entropy solution.
My talk will be separated in three parts. In a first one I will introduce the notion of stochastic entropy solution for (1) as well as a generalized notion of solution (namely measure-valued entropy solution). In a second one, I will present the monotone finite volume scheme used to approximate (1). After deriving stability estimates satisfied by the finite volume approximation (uT ; k), I will present its convergence towards a measure-valued entropy solution. The last part will be devoted to show the uniqueness of this generalized solution as well as the existence and uniqueness of the stochastic entropy solution for (1). The idea is to adapt the Kruzkhov doubling variable technique to the stochastic case by comparing any generalized solution to the finite volume approximation, having in mind the work of [BVW12].

References :
[BCC20] C. Bauzet, V. Castel, and J. Charrier. Existence and uniqueness result for an hyperbolic scalar conservation law with a stochastic force using a finite volume approximation. To appear in Journal of Hyperbolic Differential Equations.
[BCG16] C. Bauzet, J. Charrier, and T. Gallouët. Convergence of monotone finite volume schemes for hyperbolic scalar conservation laws with a multiplicative noise. Stochastic Partial Differential Equations : Analysis and Computations, Volume 4, 150-223, 2016.
[BVW12] C. Bauzet, G. Vallet, and P. Wittbold. The Cauchy problem for a conservation law with a multiplicative stochastic perturbation Journal of Hyperbolic Differential Equations, Volume 9, Issue 4, 661-709, 2012.
[DeV10] A. Debussche and J. Vovelle. Scalar conservation laws with stochastic forcing. J. Funct. Anal., 259(4):1014-1042, 2010.
[M18] A. Majee. Convergence of a flux-splitting finite volume scheme for conservation laws driven by Lévy noise. Applied Mathematics and Computation, 338(1) : 676-697, 2018.

Pendant cette période de retour progressif au travail en présentiel, le LMA organise une série de séminaires en visioconférence.

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